MAT 539 Algebraic Topology, Fall 2010.

The goal of this course is to provide a basic introduction into singular homology and cohomology.
  • Prerequisites : MAT 530.
  • References :
    Allen Hatcher, Algebraic Topology. This book is available for free download from Hatcher's webpage.
    James R. Munkres, Elements of Algebraic Topology.

  • Homework: there will be five or six problem sets, assigned bi-weekly or so. They will be posted on this page. Your course grade will be determined by homeworks.

    Important: Please write up your solutions neatly, be sure to put your name on them and staple all pages. Illegible homework will not be graded. Late homework will not be accepted. You are welcome to collaborate with others and even to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed.

    Homework 1 (pdf), due Sept 22. Please also learn about the long exact sequence of a triple, Hatcher p.118. Also read Prop. 2.19 on the same page.

    Homework 2 (pdf), due Oct 6.

    Homework 3 (pdf), due Monday, Oct 25.

    Homework 4 (pdf), due Monday, Nov 22.

    Homework 5 (pdf), due by Monday, Dec 13 (bring it to my office anytime on on before Dec 13)
  • Syllabus: I hope to cover the topics listed below, perhaps in a slightly different order. Additional topics may be discussed if time permits.
    1. Basic constructions
      • Homotopies & homotopy equivalences, homotopy type; retractions, deformation retractions
      • CW-complexes, definition, examples; simplicial complexes
      • Operations on spaces: products, quotients, wedge sums, suspension, etc
    2. Homology
      • Singular homology & simplicial homology, constructions
      • Chain complexes, chain maps, chain homotopies; exact sequences; the Euler characteristic
      • Homotopy invariance of singular homology
      • Relative homology, long exact sequence of a pair
      • The excision theorem
      • Equivalence of simplicial and singular homology
      • Homology of CW-complexes via cellular homology
      • Computations: surfaces, spheres, projective spaces, lens spaces, etc
      • Mayer-Vietoris sequence, more calculations
      • Applications: Brouwer fixed point thm, degrees, Jordan curve thm, invariance of domain, etc
      • Eilenberg-Steenrod axioms
    3. Cohomology
      • Simplicial and singular cohomology groups
      • Hom and Ext, universal coefficient theorems
      • Relative cohomology, exact sequences, isomorphism between simplicial and singular cohomology
      • Cup product, calculations (cohomology ring of projective spaces, etc)
      • Kunneth formulas
      • Poincare duality

    Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.